LMFDB - Embedded newform 98.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [98,4,Mod(1,98)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("98.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	98.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.78218718056$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	98.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-2.00000 q^{2} -1.00000 q^{3} +4.00000 q^{4} +7.00000 q^{5} +2.00000 q^{6} -8.00000 q^{8} -26.0000 q^{9} -14.0000 q^{10} +35.0000 q^{11} -4.00000 q^{12} +66.0000 q^{13} -7.00000 q^{15} +16.0000 q^{16} +59.0000 q^{17} +52.0000 q^{18} +137.000 q^{19} +28.0000 q^{20} -70.0000 q^{22} -7.00000 q^{23} +8.00000 q^{24} -76.0000 q^{25} -132.000 q^{26} +53.0000 q^{27} +106.000 q^{29} +14.0000 q^{30} +75.0000 q^{31} -32.0000 q^{32} -35.0000 q^{33} -118.000 q^{34} -104.000 q^{36} +11.0000 q^{37} -274.000 q^{38} -66.0000 q^{39} -56.0000 q^{40} -498.000 q^{41} +260.000 q^{43} +140.000 q^{44} -182.000 q^{45} +14.0000 q^{46} -171.000 q^{47} -16.0000 q^{48} +152.000 q^{50} -59.0000 q^{51} +264.000 q^{52} -417.000 q^{53} -106.000 q^{54} +245.000 q^{55} -137.000 q^{57} -212.000 q^{58} -17.0000 q^{59} -28.0000 q^{60} +51.0000 q^{61} -150.000 q^{62} +64.0000 q^{64} +462.000 q^{65} +70.0000 q^{66} +439.000 q^{67} +236.000 q^{68} +7.00000 q^{69} -784.000 q^{71} +208.000 q^{72} +295.000 q^{73} -22.0000 q^{74} +76.0000 q^{75} +548.000 q^{76} +132.000 q^{78} -495.000 q^{79} +112.000 q^{80} +649.000 q^{81} +996.000 q^{82} +932.000 q^{83} +413.000 q^{85} -520.000 q^{86} -106.000 q^{87} -280.000 q^{88} -873.000 q^{89} +364.000 q^{90} -28.0000 q^{92} -75.0000 q^{93} +342.000 q^{94} +959.000 q^{95} +32.0000 q^{96} -290.000 q^{97} -910.000 q^{99} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−1.00000	−0.192450	−0.0962250	−	0.995360i	$-0.530677\pi$
$3$	−1.00000	−0.192450	−0.0962250	+	0.995360i	$0.530677\pi$
$4$	4.00000	0.500000
$5$	7.00000	0.626099	0.313050	−	0.949737i	$-0.398649\pi$
$5$	7.00000	0.626099	0.313050	+	0.949737i	$0.398649\pi$
$6$	2.00000	0.136083
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	−26.0000	−0.962963
$10$	−14.0000	−0.442719
$11$	35.0000	0.959354	0.479677	−	0.877445i	$-0.340754\pi$
$11$	35.0000	0.959354	0.479677	+	0.877445i	$0.340754\pi$
$12$	−4.00000	−0.0962250
$13$	66.0000	1.40809	0.704043	−	0.710158i	$-0.251376\pi$
$13$	66.0000	1.40809	0.704043	+	0.710158i	$0.251376\pi$
$14$	0	0
$15$	−7.00000	−0.120493
$16$	16.0000	0.250000
$17$	59.0000	0.841741	0.420871	−	0.907121i	$-0.361725\pi$
$17$	59.0000	0.841741	0.420871	+	0.907121i	$0.361725\pi$
$18$	52.0000	0.680918
$19$	137.000	1.65421	0.827104	−	0.562049i	$-0.189987\pi$
$19$	137.000	1.65421	0.827104	+	0.562049i	$0.189987\pi$
$20$	28.0000	0.313050
$21$	0	0
$22$	−70.0000	−0.678366
$23$	−7.00000	−0.0634609	−0.0317305	−	0.999496i	$-0.510102\pi$
$23$	−7.00000	−0.0634609	−0.0317305	+	0.999496i	$0.510102\pi$
$24$	8.00000	0.0680414
$25$	−76.0000	−0.608000
$26$	−132.000	−0.995667
$27$	53.0000	0.377772
$28$	0	0
$29$	106.000	0.678748	0.339374	−	0.940651i	$-0.389785\pi$
$29$	106.000	0.678748	0.339374	+	0.940651i	$0.389785\pi$
$30$	14.0000	0.0852013
$31$	75.0000	0.434529	0.217264	−	0.976113i	$-0.430287\pi$
$31$	75.0000	0.434529	0.217264	+	0.976113i	$0.430287\pi$
$32$	−32.0000	−0.176777
$33$	−35.0000	−0.184628
$34$	−118.000	−0.595201
$35$	0	0
$36$	−104.000	−0.481481
$37$	11.0000	0.0488754	0.0244377	−	0.999701i	$-0.492220\pi$
$37$	11.0000	0.0488754	0.0244377	+	0.999701i	$0.492220\pi$
$38$	−274.000	−1.16970
$39$	−66.0000	−0.270986
$40$	−56.0000	−0.221359
$41$	−498.000	−1.89694	−0.948470	−	0.316867i	$-0.897369\pi$
$41$	−498.000	−1.89694	−0.948470	+	0.316867i	$0.897369\pi$
$42$	0	0
$43$	260.000	0.922084	0.461042	−	0.887378i	$-0.347476\pi$
$43$	260.000	0.922084	0.461042	+	0.887378i	$0.347476\pi$
$44$	140.000	0.479677
$45$	−182.000	−0.602910
$46$	14.0000	0.0448736
$47$	−171.000	−0.530700	−0.265350	−	0.964152i	$-0.585488\pi$
$47$	−171.000	−0.530700	−0.265350	+	0.964152i	$0.585488\pi$
$48$	−16.0000	−0.0481125
$49$	0	0
$50$	152.000	0.429921
$51$	−59.0000	−0.161993
$52$	264.000	0.704043
$53$	−417.000	−1.08074	−0.540371	−	0.841427i	$-0.681716\pi$
$53$	−417.000	−1.08074	−0.540371	+	0.841427i	$0.681716\pi$
$54$	−106.000	−0.267125
$55$	245.000	0.600651
$56$	0	0
$57$	−137.000	−0.318353
$58$	−212.000	−0.479948
$59$	−17.0000	−0.0375121	−0.0187560	−	0.999824i	$-0.505971\pi$
$59$	−17.0000	−0.0375121	−0.0187560	+	0.999824i	$0.505971\pi$
$60$	−28.0000	−0.0602464
$61$	51.0000	0.107047	0.0535236	−	0.998567i	$-0.482955\pi$
$61$	51.0000	0.107047	0.0535236	+	0.998567i	$0.482955\pi$
$62$	−150.000	−0.307258
$63$	0	0
$64$	64.0000	0.125000
$65$	462.000	0.881601
$66$	70.0000	0.130552
$67$	439.000	0.800483	0.400242	−	0.916410i	$-0.368926\pi$
$67$	439.000	0.800483	0.400242	+	0.916410i	$0.368926\pi$
$68$	236.000	0.420871
$69$	7.00000	0.0122131
$70$	0	0
$71$	−784.000	−1.31047	−0.655237	−	0.755423i	$-0.727431\pi$
$71$	−784.000	−1.31047	−0.655237	+	0.755423i	$0.727431\pi$
$72$	208.000	0.340459
$73$	295.000	0.472974	0.236487	−	0.971635i	$-0.424004\pi$
$73$	295.000	0.472974	0.236487	+	0.971635i	$0.424004\pi$
$74$	−22.0000	−0.0345601
$75$	76.0000	0.117010
$76$	548.000	0.827104
$77$	0	0
$78$	132.000	0.191616
$79$	−495.000	−0.704960	−0.352480	−	0.935819i	$-0.614662\pi$
$79$	−495.000	−0.704960	−0.352480	+	0.935819i	$0.614662\pi$
$80$	112.000	0.156525
$81$	649.000	0.890261
$82$	996.000	1.34134
$83$	932.000	1.23253	0.616267	−	0.787537i	$-0.288644\pi$
$83$	932.000	1.23253	0.616267	+	0.787537i	$0.288644\pi$
$84$	0	0
$85$	413.000	0.527013
$86$	−520.000	−0.652012
$87$	−106.000	−0.130625
$88$	−280.000	−0.339183
$89$	−873.000	−1.03975	−0.519875	−	0.854242i	$-0.674022\pi$
$89$	−873.000	−1.03975	−0.519875	+	0.854242i	$0.674022\pi$
$90$	364.000	0.426322
$91$	0	0
$92$	−28.0000	−0.0317305
$93$	−75.0000	−0.0836251
$94$	342.000	0.375262
$95$	959.000	1.03570
$96$	32.0000	0.0340207
$97$	−290.000	−0.303557	−0.151779	−	0.988415i	$-0.548500\pi$
$97$	−290.000	−0.303557	−0.151779	+	0.988415i	$0.548500\pi$
$98$	0	0
$99$	−910.000	−0.923823
$100$	−304.000	−0.304000
$101$	−1085.00	−1.06893	−0.534463	−	0.845192i	$-0.679486\pi$
$101$	−1085.00	−1.06893	−0.534463	+	0.845192i	$0.679486\pi$
$102$	118.000	0.114546
$103$	1553.00	1.48565	0.742823	−	0.669487i	$-0.233486\pi$
$103$	1553.00	1.48565	0.742823	+	0.669487i	$0.233486\pi$
$104$	−528.000	−0.497833
$105$	0	0
$106$	834.000	0.764200
$107$	129.000	0.116550	0.0582752	−	0.998301i	$-0.481440\pi$
$107$	129.000	0.116550	0.0582752	+	0.998301i	$0.481440\pi$
$108$	212.000	0.188886
$109$	−965.000	−0.847984	−0.423992	−	0.905666i	$-0.639372\pi$
$109$	−965.000	−0.847984	−0.423992	+	0.905666i	$0.639372\pi$
$110$	−490.000	−0.424724
$111$	−11.0000	−0.00940607
$112$	0	0
$113$	−50.0000	−0.0416248	−0.0208124	−	0.999783i	$-0.506625\pi$
$113$	−50.0000	−0.0416248	−0.0208124	+	0.999783i	$0.506625\pi$
$114$	274.000	0.225109
$115$	−49.0000	−0.0397328
$116$	424.000	0.339374
$117$	−1716.00	−1.35593
$118$	34.0000	0.0265250
$119$	0	0
$120$	56.0000	0.0426006
$121$	−106.000	−0.0796394
$122$	−102.000	−0.0756938
$123$	498.000	0.365066
$124$	300.000	0.217264
$125$	−1407.00	−1.00677
$126$	0	0
$127$	936.000	0.653989	0.326994	−	0.945026i	$-0.393964\pi$
$127$	936.000	0.653989	0.326994	+	0.945026i	$0.393964\pi$
$128$	−128.000	−0.0883883
$129$	−260.000	−0.177455
$130$	−924.000	−0.623386
$131$	−755.000	−0.503547	−0.251773	−	0.967786i	$-0.581014\pi$
$131$	−755.000	−0.503547	−0.251773	+	0.967786i	$0.581014\pi$
$132$	−140.000	−0.0923139
$133$	0	0
$134$	−878.000	−0.566027
$135$	371.000	0.236523
$136$	−472.000	−0.297600
$137$	−2357.00	−1.46987	−0.734935	−	0.678138i	$-0.762787\pi$
$137$	−2357.00	−1.46987	−0.734935	+	0.678138i	$0.762787\pi$
$138$	−14.0000	−0.00863594
$139$	28.0000	0.0170858	0.00854291	−	0.999964i	$-0.497281\pi$
$139$	28.0000	0.0170858	0.00854291	+	0.999964i	$0.497281\pi$
$140$	0	0
$141$	171.000	0.102133
$142$	1568.00	0.926645
$143$	2310.00	1.35085
$144$	−416.000	−0.240741
$145$	742.000	0.424964
$146$	−590.000	−0.334443
$147$	0	0
$148$	44.0000	0.0244377
$149$	2295.00	1.26184	0.630919	−	0.775849i	$-0.282678\pi$
$149$	2295.00	1.26184	0.630919	+	0.775849i	$0.282678\pi$
$150$	−152.000	−0.0827383
$151$	−1109.00	−0.597676	−0.298838	−	0.954304i	$-0.596599\pi$
$151$	−1109.00	−0.597676	−0.298838	+	0.954304i	$0.596599\pi$
$152$	−1096.00	−0.584851
$153$	−1534.00	−0.810566
$154$	0	0
$155$	525.000	0.272058
$156$	−264.000	−0.135493
$157$	1559.00	0.792495	0.396248	−	0.918144i	$-0.370312\pi$
$157$	1559.00	0.792495	0.396248	+	0.918144i	$0.370312\pi$
$158$	990.000	0.498482
$159$	417.000	0.207989
$160$	−224.000	−0.110680
$161$	0	0
$162$	−1298.00	−0.629509
$163$	−2251.00	−1.08167	−0.540834	−	0.841129i	$-0.681891\pi$
$163$	−2251.00	−1.08167	−0.540834	+	0.841129i	$0.681891\pi$
$164$	−1992.00	−0.948470
$165$	−245.000	−0.115595
$166$	−1864.00	−0.871533
$167$	2788.00	1.29187	0.645934	−	0.763393i	$-0.276468\pi$
$167$	2788.00	1.29187	0.645934	+	0.763393i	$0.276468\pi$
$168$	0	0
$169$	2159.00	0.982704
$170$	−826.000	−0.372655
$171$	−3562.00	−1.59294
$172$	1040.00	0.461042
$173$	1579.00	0.693926	0.346963	−	0.937879i	$-0.387213\pi$
$173$	1579.00	0.693926	0.346963	+	0.937879i	$0.387213\pi$
$174$	212.000	0.0923660
$175$	0	0
$176$	560.000	0.239839
$177$	17.0000	0.00721920
$178$	1746.00	0.735215
$179$	2451.00	1.02344	0.511722	−	0.859151i	$-0.329008\pi$
$179$	2451.00	1.02344	0.511722	+	0.859151i	$0.329008\pi$
$180$	−728.000	−0.301455
$181$	−1170.00	−0.480472	−0.240236	−	0.970715i	$-0.577225\pi$
$181$	−1170.00	−0.480472	−0.240236	+	0.970715i	$0.577225\pi$
$182$	0	0
$183$	−51.0000	−0.0206012
$184$	56.0000	0.0224368
$185$	77.0000	0.0306008
$186$	150.000	0.0591319
$187$	2065.00	0.807528
$188$	−684.000	−0.265350
$189$	0	0
$190$	−1918.00	−0.732349
$191$	−1275.00	−0.483014	−0.241507	−	0.970399i	$-0.577642\pi$
$191$	−1275.00	−0.483014	−0.241507	+	0.970399i	$0.577642\pi$
$192$	−64.0000	−0.0240563
$193$	35.0000	0.0130537	0.00652683	−	0.999979i	$-0.497922\pi$
$193$	35.0000	0.0130537	0.00652683	+	0.999979i	$0.497922\pi$
$194$	580.000	0.214647
$195$	−462.000	−0.169664
$196$	0	0
$197$	−2734.00	−0.988779	−0.494389	−	0.869241i	$-0.664608\pi$
$197$	−2734.00	−0.988779	−0.494389	+	0.869241i	$0.664608\pi$
$198$	1820.00	0.653241
$199$	2243.00	0.799005	0.399503	−	0.916732i	$-0.369183\pi$
$199$	2243.00	0.799005	0.399503	+	0.916732i	$0.369183\pi$
$200$	608.000	0.214960
$201$	−439.000	−0.154053
$202$	2170.00	0.755845
$203$	0	0
$204$	−236.000	−0.0809966
$205$	−3486.00	−1.18767
$206$	−3106.00	−1.05051
$207$	182.000	0.0611105
$208$	1056.00	0.352021
$209$	4795.00	1.58697
$210$	0	0
$211$	1172.00	0.382388	0.191194	−	0.981552i	$-0.438764\pi$
$211$	1172.00	0.382388	0.191194	+	0.981552i	$0.438764\pi$
$212$	−1668.00	−0.540371
$213$	784.000	0.252201
$214$	−258.000	−0.0824136
$215$	1820.00	0.577316
$216$	−424.000	−0.133563
$217$	0	0
$218$	1930.00	0.599615
$219$	−295.000	−0.0910240
$220$	980.000	0.300325
$221$	3894.00	1.18524
$222$	22.0000	0.00665110
$223$	2024.00	0.607790	0.303895	−	0.952706i	$-0.401713\pi$
$223$	2024.00	0.607790	0.303895	+	0.952706i	$0.401713\pi$
$224$	0	0
$225$	1976.00	0.585481
$226$	100.000	0.0294332
$227$	2571.00	0.751732	0.375866	−	0.926674i	$-0.377345\pi$
$227$	2571.00	0.751732	0.375866	+	0.926674i	$0.377345\pi$
$228$	−548.000	−0.159176
$229$	895.000	0.258268	0.129134	−	0.991627i	$-0.458780\pi$
$229$	895.000	0.258268	0.129134	+	0.991627i	$0.458780\pi$
$230$	98.0000	0.0280953
$231$	0	0
$232$	−848.000	−0.239974
$233$	1787.00	0.502447	0.251224	−	0.967929i	$-0.419167\pi$
$233$	1787.00	0.502447	0.251224	+	0.967929i	$0.419167\pi$
$234$	3432.00	0.958790
$235$	−1197.00	−0.332271
$236$	−68.0000	−0.0187560
$237$	495.000	0.135670
$238$	0	0
$239$	−5100.00	−1.38030	−0.690150	−	0.723667i	$-0.742455\pi$
$239$	−5100.00	−1.38030	−0.690150	+	0.723667i	$0.742455\pi$
$240$	−112.000	−0.0301232
$241$	−4177.00	−1.11645	−0.558225	−	0.829690i	$-0.688517\pi$
$241$	−4177.00	−1.11645	−0.558225	+	0.829690i	$0.688517\pi$
$242$	212.000	0.0563135
$243$	−2080.00	−0.549103
$244$	204.000	0.0535236
$245$	0	0
$246$	−996.000	−0.258141
$247$	9042.00	2.32927
$248$	−600.000	−0.153629
$249$	−932.000	−0.237201
$250$	2814.00	0.711892
$251$	−4680.00	−1.17689	−0.588444	−	0.808538i	$-0.700259\pi$
$251$	−4680.00	−1.17689	−0.588444	+	0.808538i	$0.700259\pi$
$252$	0	0
$253$	−245.000	−0.0608815
$254$	−1872.00	−0.462440
$255$	−413.000	−0.101424
$256$	256.000	0.0625000
$257$	−1749.00	−0.424512	−0.212256	−	0.977214i	$-0.568081\pi$
$257$	−1749.00	−0.424512	−0.212256	+	0.977214i	$0.568081\pi$
$258$	520.000	0.125480
$259$	0	0
$260$	1848.00	0.440800
$261$	−2756.00	−0.653610
$262$	1510.00	0.356061
$263$	−4473.00	−1.04873	−0.524367	−	0.851492i	$-0.675698\pi$
$263$	−4473.00	−1.04873	−0.524367	+	0.851492i	$0.675698\pi$
$264$	280.000	0.0652758
$265$	−2919.00	−0.676652
$266$	0	0
$267$	873.000	0.200100
$268$	1756.00	0.400242
$269$	1975.00	0.447650	0.223825	−	0.974629i	$-0.428146\pi$
$269$	1975.00	0.447650	0.223825	+	0.974629i	$0.428146\pi$
$270$	−742.000	−0.167247
$271$	−8439.00	−1.89163	−0.945817	−	0.324701i	$-0.894736\pi$
$271$	−8439.00	−1.89163	−0.945817	+	0.324701i	$0.894736\pi$
$272$	944.000	0.210435
$273$	0	0
$274$	4714.00	1.03935
$275$	−2660.00	−0.583287
$276$	28.0000	0.00610653
$277$	527.000	0.114312	0.0571559	−	0.998365i	$-0.481797\pi$
$277$	527.000	0.114312	0.0571559	+	0.998365i	$0.481797\pi$
$278$	−56.0000	−0.0120815
$279$	−1950.00	−0.418435
$280$	0	0
$281$	−202.000	−0.0428837	−0.0214418	−	0.999770i	$-0.506826\pi$
$281$	−202.000	−0.0428837	−0.0214418	+	0.999770i	$0.506826\pi$
$282$	−342.000	−0.0722192
$283$	−7949.00	−1.66968	−0.834839	−	0.550494i	$-0.814439\pi$
$283$	−7949.00	−1.66968	−0.834839	+	0.550494i	$0.814439\pi$
$284$	−3136.00	−0.655237
$285$	−959.000	−0.199320
$286$	−4620.00	−0.955197
$287$	0	0
$288$	832.000	0.170229
$289$	−1432.00	−0.291472
$290$	−1484.00	−0.300495
$291$	290.000	0.0584196
$292$	1180.00	0.236487
$293$	318.000	0.0634053	0.0317027	−	0.999497i	$-0.489907\pi$
$293$	318.000	0.0634053	0.0317027	+	0.999497i	$0.489907\pi$
$294$	0	0
$295$	−119.000	−0.0234863
$296$	−88.0000	−0.0172801
$297$	1855.00	0.362418
$298$	−4590.00	−0.892254
$299$	−462.000	−0.0893584
$300$	304.000	0.0585048
$301$	0	0
$302$	2218.00	0.422621
$303$	1085.00	0.205715
$304$	2192.00	0.413552
$305$	357.000	0.0670222
$306$	3068.00	0.573156
$307$	−8132.00	−1.51178	−0.755892	−	0.654696i	$-0.772797\pi$
$307$	−8132.00	−1.51178	−0.755892	+	0.654696i	$0.772797\pi$
$308$	0	0
$309$	−1553.00	−0.285913
$310$	−1050.00	−0.192374
$311$	−929.000	−0.169385	−0.0846925	−	0.996407i	$-0.526991\pi$
$311$	−929.000	−0.169385	−0.0846925	+	0.996407i	$0.526991\pi$
$312$	528.000	0.0958081
$313$	−209.000	−0.0377424	−0.0188712	−	0.999822i	$-0.506007\pi$
$313$	−209.000	−0.0377424	−0.0188712	+	0.999822i	$0.506007\pi$
$314$	−3118.00	−0.560379
$315$	0	0
$316$	−1980.00	−0.352480
$317$	7131.00	1.26346	0.631730	−	0.775188i	$-0.282345\pi$
$317$	7131.00	1.26346	0.631730	+	0.775188i	$0.282345\pi$
$318$	−834.000	−0.147070
$319$	3710.00	0.651160
$320$	448.000	0.0782624
$321$	−129.000	−0.0224301
$322$	0	0
$323$	8083.00	1.39242
$324$	2596.00	0.445130
$325$	−5016.00	−0.856116
$326$	4502.00	0.764855
$327$	965.000	0.163195
$328$	3984.00	0.670670
$329$	0	0
$330$	490.000	0.0817382
$331$	−6571.00	−1.09116	−0.545581	−	0.838058i	$-0.683691\pi$
$331$	−6571.00	−1.09116	−0.545581	+	0.838058i	$0.683691\pi$
$332$	3728.00	0.616267
$333$	−286.000	−0.0470652
$334$	−5576.00	−0.913488
$335$	3073.00	0.501182
$336$	0	0
$337$	−11466.0	−1.85339	−0.926696	−	0.375813i	$-0.877364\pi$
$337$	−11466.0	−1.85339	−0.926696	+	0.375813i	$0.877364\pi$
$338$	−4318.00	−0.694876
$339$	50.0000	0.00801070
$340$	1652.00	0.263507
$341$	2625.00	0.416867
$342$	7124.00	1.12638
$343$	0	0
$344$	−2080.00	−0.326006
$345$	49.0000	0.00764658
$346$	−3158.00	−0.490680
$347$	−9777.00	−1.51256	−0.756278	−	0.654251i	$-0.772984\pi$
$347$	−9777.00	−1.51256	−0.756278	+	0.654251i	$0.772984\pi$
$348$	−424.000	−0.0653126
$349$	11914.0	1.82734	0.913670	−	0.406456i	$-0.133236\pi$
$349$	11914.0	1.82734	0.913670	+	0.406456i	$0.133236\pi$
$350$	0	0
$351$	3498.00	0.531936
$352$	−1120.00	−0.169591
$353$	9123.00	1.37555	0.687774	−	0.725925i	$-0.258588\pi$
$353$	9123.00	1.37555	0.687774	+	0.725925i	$0.258588\pi$
$354$	−34.0000	−0.00510474
$355$	−5488.00	−0.820487
$356$	−3492.00	−0.519875
$357$	0	0
$358$	−4902.00	−0.723684
$359$	8149.00	1.19802	0.599008	−	0.800743i	$-0.295562\pi$
$359$	8149.00	1.19802	0.599008	+	0.800743i	$0.295562\pi$
$360$	1456.00	0.213161
$361$	11910.0	1.73640
$362$	2340.00	0.339745
$363$	106.000	0.0153266
$364$	0	0
$365$	2065.00	0.296129
$366$	102.000	0.0145673
$367$	9671.00	1.37554	0.687769	−	0.725930i	$-0.258590\pi$
$367$	9671.00	1.37554	0.687769	+	0.725930i	$0.258590\pi$
$368$	−112.000	−0.0158652
$369$	12948.0	1.82668
$370$	−154.000	−0.0216381
$371$	0	0
$372$	−300.000	−0.0418126
$373$	−4109.00	−0.570391	−0.285196	−	0.958469i	$-0.592059\pi$
$373$	−4109.00	−0.570391	−0.285196	+	0.958469i	$0.592059\pi$
$374$	−4130.00	−0.571009
$375$	1407.00	0.193752
$376$	1368.00	0.187631
$377$	6996.00	0.955736
$378$	0	0
$379$	−3488.00	−0.472735	−0.236367	−	0.971664i	$-0.575957\pi$
$379$	−3488.00	−0.472735	−0.236367	+	0.971664i	$0.575957\pi$
$380$	3836.00	0.517849
$381$	−936.000	−0.125860
$382$	2550.00	0.341543
$383$	8717.00	1.16297	0.581485	−	0.813557i	$-0.302472\pi$
$383$	8717.00	1.16297	0.581485	+	0.813557i	$0.302472\pi$
$384$	128.000	0.0170103
$385$	0	0
$386$	−70.0000	−0.00923033
$387$	−6760.00	−0.887933
$388$	−1160.00	−0.151779
$389$	163.000	0.0212453	0.0106227	−	0.999944i	$-0.496619\pi$
$389$	163.000	0.0212453	0.0106227	+	0.999944i	$0.496619\pi$
$390$	924.000	0.119971
$391$	−413.000	−0.0534177
$392$	0	0
$393$	755.000	0.0969077
$394$	5468.00	0.699172
$395$	−3465.00	−0.441375
$396$	−3640.00	−0.461911
$397$	999.000	0.126293	0.0631466	−	0.998004i	$-0.479886\pi$
$397$	999.000	0.126293	0.0631466	+	0.998004i	$0.479886\pi$
$398$	−4486.00	−0.564982
$399$	0	0
$400$	−1216.00	−0.152000
$401$	−14757.0	−1.83773	−0.918865	−	0.394573i	$-0.870893\pi$
$401$	−14757.0	−1.83773	−0.918865	+	0.394573i	$0.870893\pi$
$402$	878.000	0.108932
$403$	4950.00	0.611854
$404$	−4340.00	−0.534463
$405$	4543.00	0.557391
$406$	0	0
$407$	385.000	0.0468888
$408$	472.000	0.0572732
$409$	−133.000	−0.0160793	−0.00803964	−	0.999968i	$-0.502559\pi$
$409$	−133.000	−0.0160793	−0.00803964	+	0.999968i	$0.502559\pi$
$410$	6972.00	0.839811
$411$	2357.00	0.282876
$412$	6212.00	0.742823
$413$	0	0
$414$	−364.000	−0.0432117
$415$	6524.00	0.771688
$416$	−2112.00	−0.248917
$417$	−28.0000	−0.00328817
$418$	−9590.00	−1.12216
$419$	−6420.00	−0.748538	−0.374269	−	0.927320i	$-0.622106\pi$
$419$	−6420.00	−0.748538	−0.374269	+	0.927320i	$0.622106\pi$
$420$	0	0
$421$	10266.0	1.18844	0.594221	−	0.804302i	$-0.297460\pi$
$421$	10266.0	1.18844	0.594221	+	0.804302i	$0.297460\pi$
$422$	−2344.00	−0.270389
$423$	4446.00	0.511045
$424$	3336.00	0.382100
$425$	−4484.00	−0.511779
$426$	−1568.00	−0.178333
$427$	0	0
$428$	516.000	0.0582752
$429$	−2310.00	−0.259972
$430$	−3640.00	−0.408224
$431$	−15213.0	−1.70020	−0.850098	−	0.526625i	$-0.823457\pi$
$431$	−15213.0	−1.70020	−0.850098	+	0.526625i	$0.823457\pi$
$432$	848.000	0.0944431
$433$	−1378.00	−0.152939	−0.0764693	−	0.997072i	$-0.524365\pi$
$433$	−1378.00	−0.152939	−0.0764693	+	0.997072i	$0.524365\pi$
$434$	0	0
$435$	−742.000	−0.0817843
$436$	−3860.00	−0.423992
$437$	−959.000	−0.104978
$438$	590.000	0.0643637
$439$	−2763.00	−0.300389	−0.150195	−	0.988656i	$-0.547990\pi$
$439$	−2763.00	−0.300389	−0.150195	+	0.988656i	$0.547990\pi$
$440$	−1960.00	−0.212362
$441$	0	0
$442$	−7788.00	−0.838094
$443$	5849.00	0.627301	0.313651	−	0.949538i	$-0.398448\pi$
$443$	5849.00	0.627301	0.313651	+	0.949538i	$0.398448\pi$
$444$	−44.0000	−0.00470304
$445$	−6111.00	−0.650987
$446$	−4048.00	−0.429772
$447$	−2295.00	−0.242841
$448$	0	0
$449$	4582.00	0.481599	0.240799	−	0.970575i	$-0.422590\pi$
$449$	4582.00	0.481599	0.240799	+	0.970575i	$0.422590\pi$
$450$	−3952.00	−0.413998
$451$	−17430.0	−1.81984
$452$	−200.000	−0.0208124
$453$	1109.00	0.115023
$454$	−5142.00	−0.531555
$455$	0	0
$456$	1096.00	0.112555
$457$	11551.0	1.18235	0.591174	−	0.806544i	$-0.298665\pi$
$457$	11551.0	1.18235	0.591174	+	0.806544i	$0.298665\pi$
$458$	−1790.00	−0.182623
$459$	3127.00	0.317987
$460$	−196.000	−0.0198664
$461$	−9494.00	−0.959175	−0.479587	−	0.877494i	$-0.659214\pi$
$461$	−9494.00	−0.959175	−0.479587	+	0.877494i	$0.659214\pi$
$462$	0	0
$463$	−10160.0	−1.01982	−0.509908	−	0.860229i	$-0.670321\pi$
$463$	−10160.0	−1.01982	−0.509908	+	0.860229i	$0.670321\pi$
$464$	1696.00	0.169687
$465$	−525.000	−0.0523576
$466$	−3574.00	−0.355284
$467$	−1307.00	−0.129509	−0.0647545	−	0.997901i	$-0.520626\pi$
$467$	−1307.00	−0.129509	−0.0647545	+	0.997901i	$0.520626\pi$
$468$	−6864.00	−0.677967
$469$	0	0
$470$	2394.00	0.234951
$471$	−1559.00	−0.152516
$472$	136.000	0.0132625
$473$	9100.00	0.884606
$474$	−990.000	−0.0959329
$475$	−10412.0	−1.00576
$476$	0	0
$477$	10842.0	1.04072
$478$	10200.0	0.976019
$479$	18287.0	1.74437	0.872186	−	0.489174i	$-0.162702\pi$
$479$	18287.0	1.74437	0.872186	+	0.489174i	$0.162702\pi$
$480$	224.000	0.0213003
$481$	726.000	0.0688207
$482$	8354.00	0.789449
$483$	0	0
$484$	−424.000	−0.0398197
$485$	−2030.00	−0.190057
$486$	4160.00	0.388275
$487$	−14953.0	−1.39135	−0.695673	−	0.718359i	$-0.744894\pi$
$487$	−14953.0	−1.39135	−0.695673	+	0.718359i	$0.744894\pi$
$488$	−408.000	−0.0378469
$489$	2251.00	0.208167
$490$	0	0
$491$	14352.0	1.31914	0.659569	−	0.751644i	$-0.270739\pi$
$491$	14352.0	1.31914	0.659569	+	0.751644i	$0.270739\pi$
$492$	1992.00	0.182533
$493$	6254.00	0.571331
$494$	−18084.0	−1.64704
$495$	−6370.00	−0.578404
$496$	1200.00	0.108632
$497$	0	0
$498$	1864.00	0.167727
$499$	−5531.00	−0.496196	−0.248098	−	0.968735i	$-0.579805\pi$
$499$	−5531.00	−0.496196	−0.248098	+	0.968735i	$0.579805\pi$
$500$	−5628.00	−0.503384
$501$	−2788.00	−0.248620
$502$	9360.00	0.832186
$503$	8400.00	0.744607	0.372304	−	0.928111i	$-0.378568\pi$
$503$	8400.00	0.744607	0.372304	+	0.928111i	$0.378568\pi$
$504$	0	0
$505$	−7595.00	−0.669254
$506$	490.000	0.0430497
$507$	−2159.00	−0.189121
$508$	3744.00	0.326994
$509$	−2385.00	−0.207688	−0.103844	−	0.994594i	$-0.533114\pi$
$509$	−2385.00	−0.207688	−0.103844	+	0.994594i	$0.533114\pi$
$510$	826.000	0.0717174
$511$	0	0
$512$	−512.000	−0.0441942
$513$	7261.00	0.624914
$514$	3498.00	0.300175
$515$	10871.0	0.930162
$516$	−1040.00	−0.0887276
$517$	−5985.00	−0.509130
$518$	0	0
$519$	−1579.00	−0.133546
$520$	−3696.00	−0.311693
$521$	−9153.00	−0.769674	−0.384837	−	0.922985i	$-0.625742\pi$
$521$	−9153.00	−0.769674	−0.384837	+	0.922985i	$0.625742\pi$
$522$	5512.00	0.462172
$523$	−13807.0	−1.15437	−0.577187	−	0.816612i	$-0.695850\pi$
$523$	−13807.0	−1.15437	−0.577187	+	0.816612i	$0.695850\pi$
$524$	−3020.00	−0.251773
$525$	0	0
$526$	8946.00	0.741567
$527$	4425.00	0.365761
$528$	−560.000	−0.0461570
$529$	−12118.0	−0.995973
$530$	5838.00	0.478465
$531$	442.000	0.0361227
$532$	0	0
$533$	−32868.0	−2.67105
$534$	−1746.00	−0.141492
$535$	903.000	0.0729721
$536$	−3512.00	−0.283014
$537$	−2451.00	−0.196962
$538$	−3950.00	−0.316536
$539$	0	0
$540$	1484.00	0.118261
$541$	8175.00	0.649669	0.324834	−	0.945771i	$-0.394691\pi$
$541$	8175.00	0.649669	0.324834	+	0.945771i	$0.394691\pi$
$542$	16878.0	1.33759
$543$	1170.00	0.0924669
$544$	−1888.00	−0.148800
$545$	−6755.00	−0.530922
$546$	0	0
$547$	4656.00	0.363942	0.181971	−	0.983304i	$-0.441752\pi$
$547$	4656.00	0.363942	0.181971	+	0.983304i	$0.441752\pi$
$548$	−9428.00	−0.734935
$549$	−1326.00	−0.103083
$550$	5320.00	0.412446
$551$	14522.0	1.12279
$552$	−56.0000	−0.00431797
$553$	0	0
$554$	−1054.00	−0.0808306
$555$	−77.0000	−0.00588913
$556$	112.000	0.00854291
$557$	7003.00	0.532723	0.266361	−	0.963873i	$-0.414179\pi$
$557$	7003.00	0.532723	0.266361	+	0.963873i	$0.414179\pi$
$558$	3900.00	0.295878
$559$	17160.0	1.29837
$560$	0	0
$561$	−2065.00	−0.155409
$562$	404.000	0.0303233
$563$	−19753.0	−1.47867	−0.739334	−	0.673339i	$-0.764859\pi$
$563$	−19753.0	−1.47867	−0.739334	+	0.673339i	$0.764859\pi$
$564$	684.000	0.0510667
$565$	−350.000	−0.0260613
$566$	15898.0	1.18064
$567$	0	0
$568$	6272.00	0.463323
$569$	−6897.00	−0.508150	−0.254075	−	0.967185i	$-0.581771\pi$
$569$	−6897.00	−0.508150	−0.254075	+	0.967185i	$0.581771\pi$
$570$	1918.00	0.140941
$571$	24915.0	1.82603	0.913013	−	0.407932i	$-0.133750\pi$
$571$	24915.0	1.82603	0.913013	+	0.407932i	$0.133750\pi$
$572$	9240.00	0.675426
$573$	1275.00	0.0929562
$574$	0	0
$575$	532.000	0.0385842
$576$	−1664.00	−0.120370
$577$	127.000	0.00916305	0.00458152	−	0.999990i	$-0.498542\pi$
$577$	127.000	0.00916305	0.00458152	+	0.999990i	$0.498542\pi$
$578$	2864.00	0.206102
$579$	−35.0000	−0.00251218
$580$	2968.00	0.212482
$581$	0	0
$582$	−580.000	−0.0413089
$583$	−14595.0	−1.03681
$584$	−2360.00	−0.167222
$585$	−12012.0	−0.848949
$586$	−636.000	−0.0448343
$587$	9044.00	0.635921	0.317961	−	0.948104i	$-0.397002\pi$
$587$	9044.00	0.635921	0.317961	+	0.948104i	$0.397002\pi$
$588$	0	0
$589$	10275.0	0.718801
$590$	238.000	0.0166073
$591$	2734.00	0.190291
$592$	176.000	0.0122188
$593$	−10701.0	−0.741041	−0.370521	−	0.928824i	$-0.620821\pi$
$593$	−10701.0	−0.741041	−0.370521	+	0.928824i	$0.620821\pi$
$594$	−3710.00	−0.256268
$595$	0	0
$596$	9180.00	0.630919
$597$	−2243.00	−0.153769
$598$	924.000	0.0631859
$599$	20799.0	1.41874	0.709369	−	0.704837i	$-0.248980\pi$
$599$	20799.0	1.41874	0.709369	+	0.704837i	$0.248980\pi$
$600$	−608.000	−0.0413692
$601$	−1402.00	−0.0951560	−0.0475780	−	0.998868i	$-0.515150\pi$
$601$	−1402.00	−0.0951560	−0.0475780	+	0.998868i	$0.515150\pi$
$602$	0	0
$603$	−11414.0	−0.770836
$604$	−4436.00	−0.298838
$605$	−742.000	−0.0498621
$606$	−2170.00	−0.145462
$607$	6525.00	0.436312	0.218156	−	0.975914i	$-0.429996\pi$
$607$	6525.00	0.436312	0.218156	+	0.975914i	$0.429996\pi$
$608$	−4384.00	−0.292425
$609$	0	0
$610$	−714.000	−0.0473918
$611$	−11286.0	−0.747271
$612$	−6136.00	−0.405283
$613$	15051.0	0.991687	0.495844	−	0.868412i	$-0.334859\pi$
$613$	15051.0	0.991687	0.495844	+	0.868412i	$0.334859\pi$
$614$	16264.0	1.06899
$615$	3486.00	0.228568
$616$	0	0
$617$	11150.0	0.727524	0.363762	−	0.931492i	$-0.381492\pi$
$617$	11150.0	0.727524	0.363762	+	0.931492i	$0.381492\pi$
$618$	3106.00	0.202171
$619$	3415.00	0.221745	0.110873	−	0.993835i	$-0.464635\pi$
$619$	3415.00	0.221745	0.110873	+	0.993835i	$0.464635\pi$
$620$	2100.00	0.136029
$621$	−371.000	−0.0239738
$622$	1858.00	0.119773
$623$	0	0
$624$	−1056.00	−0.0677465
$625$	−349.000	−0.0223360
$626$	418.000	0.0266879
$627$	−4795.00	−0.305413
$628$	6236.00	0.396248
$629$	649.000	0.0411404
$630$	0	0
$631$	−21184.0	−1.33648	−0.668242	−	0.743944i	$-0.732953\pi$
$631$	−21184.0	−1.33648	−0.668242	+	0.743944i	$0.732953\pi$
$632$	3960.00	0.249241
$633$	−1172.00	−0.0735905
$634$	−14262.0	−0.893401
$635$	6552.00	0.409462
$636$	1668.00	0.103995
$637$	0	0
$638$	−7420.00	−0.460440
$639$	20384.0	1.26194
$640$	−896.000	−0.0553399
$641$	−10705.0	−0.659629	−0.329814	−	0.944046i	$-0.606986\pi$
$641$	−10705.0	−0.659629	−0.329814	+	0.944046i	$0.606986\pi$
$642$	258.000	0.0158605
$643$	6860.00	0.420734	0.210367	−	0.977622i	$-0.432534\pi$
$643$	6860.00	0.420734	0.210367	+	0.977622i	$0.432534\pi$
$644$	0	0
$645$	−1820.00	−0.111105
$646$	−16166.0	−0.984586
$647$	14463.0	0.878824	0.439412	−	0.898286i	$-0.355187\pi$
$647$	14463.0	0.878824	0.439412	+	0.898286i	$0.355187\pi$
$648$	−5192.00	−0.314755
$649$	−595.000	−0.0359874
$650$	10032.0	0.605365
$651$	0	0
$652$	−9004.00	−0.540834
$653$	5979.00	0.358310	0.179155	−	0.983821i	$-0.442664\pi$
$653$	5979.00	0.358310	0.179155	+	0.983821i	$0.442664\pi$
$654$	−1930.00	−0.115396
$655$	−5285.00	−0.315270
$656$	−7968.00	−0.474235
$657$	−7670.00	−0.455457
$658$	0	0
$659$	−6940.00	−0.410234	−0.205117	−	0.978737i	$-0.565757\pi$
$659$	−6940.00	−0.410234	−0.205117	+	0.978737i	$0.565757\pi$
$660$	−980.000	−0.0577976
$661$	13399.0	0.788443	0.394221	−	0.919015i	$-0.371014\pi$
$661$	13399.0	0.788443	0.394221	+	0.919015i	$0.371014\pi$
$662$	13142.0	0.771568
$663$	−3894.00	−0.228100
$664$	−7456.00	−0.435766
$665$	0	0
$666$	572.000	0.0332801
$667$	−742.000	−0.0430740
$668$	11152.0	0.645934
$669$	−2024.00	−0.116969
$670$	−6146.00	−0.354389
$671$	1785.00	0.102696
$672$	0	0
$673$	29510.0	1.69023	0.845117	−	0.534582i	$-0.179531\pi$
$673$	29510.0	1.69023	0.845117	+	0.534582i	$0.179531\pi$
$674$	22932.0	1.31055
$675$	−4028.00	−0.229686
$676$	8636.00	0.491352
$677$	−26001.0	−1.47607	−0.738035	−	0.674762i	$-0.764246\pi$
$677$	−26001.0	−1.47607	−0.738035	+	0.674762i	$0.764246\pi$
$678$	−100.000	−0.00566442
$679$	0	0
$680$	−3304.00	−0.186327
$681$	−2571.00	−0.144671
$682$	−5250.00	−0.294770
$683$	−8805.00	−0.493285	−0.246643	−	0.969106i	$-0.579327\pi$
$683$	−8805.00	−0.493285	−0.246643	+	0.969106i	$0.579327\pi$
$684$	−14248.0	−0.796471
$685$	−16499.0	−0.920284
$686$	0	0
$687$	−895.000	−0.0497036
$688$	4160.00	0.230521
$689$	−27522.0	−1.52178
$690$	−98.0000	−0.00540695
$691$	28685.0	1.57920	0.789601	−	0.613620i	$-0.210287\pi$
$691$	28685.0	1.57920	0.789601	+	0.613620i	$0.210287\pi$
$692$	6316.00	0.346963
$693$	0	0
$694$	19554.0	1.06954
$695$	196.000	0.0106974
$696$	848.000	0.0461830
$697$	−29382.0	−1.59673
$698$	−23828.0	−1.29212
$699$	−1787.00	−0.0966961
$700$	0	0
$701$	−3146.00	−0.169505	−0.0847523	−	0.996402i	$-0.527010\pi$
$701$	−3146.00	−0.169505	−0.0847523	+	0.996402i	$0.527010\pi$
$702$	−6996.00	−0.376135
$703$	1507.00	0.0808500
$704$	2240.00	0.119919
$705$	1197.00	0.0639456
$706$	−18246.0	−0.972659
$707$	0	0
$708$	68.0000	0.00360960
$709$	1259.00	0.0666893	0.0333447	−	0.999444i	$-0.489384\pi$
$709$	1259.00	0.0666893	0.0333447	+	0.999444i	$0.489384\pi$
$710$	10976.0	0.580172
$711$	12870.0	0.678851
$712$	6984.00	0.367607
$713$	−525.000	−0.0275756
$714$	0	0
$715$	16170.0	0.845767
$716$	9804.00	0.511722
$717$	5100.00	0.265639
$718$	−16298.0	−0.847125
$719$	16425.0	0.851946	0.425973	−	0.904736i	$-0.359932\pi$
$719$	16425.0	0.851946	0.425973	+	0.904736i	$0.359932\pi$
$720$	−2912.00	−0.150728
$721$	0	0
$722$	−23820.0	−1.22782
$723$	4177.00	0.214861
$724$	−4680.00	−0.240236
$725$	−8056.00	−0.412679
$726$	−212.000	−0.0108375
$727$	−6032.00	−0.307723	−0.153861	−	0.988092i	$-0.549171\pi$
$727$	−6032.00	−0.307723	−0.153861	+	0.988092i	$0.549171\pi$
$728$	0	0
$729$	−15443.0	−0.784586
$730$	−4130.00	−0.209395
$731$	15340.0	0.776156
$732$	−204.000	−0.0103006
$733$	15243.0	0.768094	0.384047	−	0.923314i	$-0.374530\pi$
$733$	15243.0	0.768094	0.384047	+	0.923314i	$0.374530\pi$
$734$	−19342.0	−0.972652
$735$	0	0
$736$	224.000	0.0112184
$737$	15365.0	0.767947
$738$	−25896.0	−1.29166
$739$	−10053.0	−0.500414	−0.250207	−	0.968192i	$-0.580499\pi$
$739$	−10053.0	−0.500414	−0.250207	+	0.968192i	$0.580499\pi$
$740$	308.000	0.0153004
$741$	−9042.00	−0.448267
$742$	0	0
$743$	24384.0	1.20399	0.601993	−	0.798501i	$-0.294373\pi$
$743$	24384.0	1.20399	0.601993	+	0.798501i	$0.294373\pi$
$744$	600.000	0.0295660
$745$	16065.0	0.790035
$746$	8218.00	0.403328
$747$	−24232.0	−1.18688
$748$	8260.00	0.403764
$749$	0	0
$750$	−2814.00	−0.137004
$751$	11589.0	0.563101	0.281550	−	0.959546i	$-0.409151\pi$
$751$	11589.0	0.563101	0.281550	+	0.959546i	$0.409151\pi$
$752$	−2736.00	−0.132675
$753$	4680.00	0.226492
$754$	−13992.0	−0.675807
$755$	−7763.00	−0.374205
$756$	0	0
$757$	14562.0	0.699161	0.349581	−	0.936906i	$-0.386324\pi$
$757$	14562.0	0.699161	0.349581	+	0.936906i	$0.386324\pi$
$758$	6976.00	0.334274
$759$	245.000	0.0117166
$760$	−7672.00	−0.366175
$761$	−22765.0	−1.08440	−0.542201	−	0.840249i	$-0.682409\pi$
$761$	−22765.0	−1.08440	−0.542201	+	0.840249i	$0.682409\pi$
$762$	1872.00	0.0889966
$763$	0	0
$764$	−5100.00	−0.241507
$765$	−10738.0	−0.507494
$766$	−17434.0	−0.822345
$767$	−1122.00	−0.0528202
$768$	−256.000	−0.0120281
$769$	3766.00	0.176600	0.0883000	−	0.996094i	$-0.471857\pi$
$769$	3766.00	0.176600	0.0883000	+	0.996094i	$0.471857\pi$
$770$	0	0
$771$	1749.00	0.0816974
$772$	140.000	0.00652683
$773$	−26861.0	−1.24984	−0.624918	−	0.780691i	$-0.714868\pi$
$773$	−26861.0	−1.24984	−0.624918	+	0.780691i	$0.714868\pi$
$774$	13520.0	0.627864
$775$	−5700.00	−0.264194
$776$	2320.00	0.107324
$777$	0	0
$778$	−326.000	−0.0150227
$779$	−68226.0	−3.13793
$780$	−1848.00	−0.0848321
$781$	−27440.0	−1.25721
$782$	826.000	0.0377720
$783$	5618.00	0.256412
$784$	0	0
$785$	10913.0	0.496180
$786$	−1510.00	−0.0685241
$787$	−2097.00	−0.0949809	−0.0474905	−	0.998872i	$-0.515122\pi$
$787$	−2097.00	−0.0949809	−0.0474905	+	0.998872i	$0.515122\pi$
$788$	−10936.0	−0.494389
$789$	4473.00	0.201829
$790$	6930.00	0.312099
$791$	0	0
$792$	7280.00	0.326621
$793$	3366.00	0.150732
$794$	−1998.00	−0.0893027
$795$	2919.00	0.130222
$796$	8972.00	0.399503
$797$	−35334.0	−1.57038	−0.785191	−	0.619254i	$-0.787435\pi$
$797$	−35334.0	−1.57038	−0.785191	+	0.619254i	$0.787435\pi$
$798$	0	0
$799$	−10089.0	−0.446712
$800$	2432.00	0.107480
$801$	22698.0	1.00124
$802$	29514.0	1.29947
$803$	10325.0	0.453750
$804$	−1756.00	−0.0770265
$805$	0	0
$806$	−9900.00	−0.432646
$807$	−1975.00	−0.0861503
$808$	8680.00	0.377922
$809$	42535.0	1.84852	0.924259	−	0.381766i	$-0.124684\pi$
$809$	42535.0	1.84852	0.924259	+	0.381766i	$0.124684\pi$
$810$	−9086.00	−0.394135
$811$	30676.0	1.32821	0.664106	−	0.747638i	$-0.268812\pi$
$811$	30676.0	1.32821	0.664106	+	0.747638i	$0.268812\pi$
$812$	0	0
$813$	8439.00	0.364045
$814$	−770.000	−0.0331554
$815$	−15757.0	−0.677231
$816$	−944.000	−0.0404983
$817$	35620.0	1.52532
$818$	266.000	0.0113698
$819$	0	0
$820$	−13944.0	−0.593836
$821$	37343.0	1.58743	0.793715	−	0.608290i	$-0.208144\pi$
$821$	37343.0	1.58743	0.793715	+	0.608290i	$0.208144\pi$
$822$	−4714.00	−0.200024
$823$	2815.00	0.119228	0.0596141	−	0.998222i	$-0.481013\pi$
$823$	2815.00	0.119228	0.0596141	+	0.998222i	$0.481013\pi$
$824$	−12424.0	−0.525256
$825$	2660.00	0.112254
$826$	0	0
$827$	−9276.00	−0.390034	−0.195017	−	0.980800i	$-0.562476\pi$
$827$	−9276.00	−0.390034	−0.195017	+	0.980800i	$0.562476\pi$
$828$	728.000	0.0305553
$829$	18571.0	0.778043	0.389021	−	0.921229i	$-0.372813\pi$
$829$	18571.0	0.778043	0.389021	+	0.921229i	$0.372813\pi$
$830$	−13048.0	−0.545666
$831$	−527.000	−0.0219993
$832$	4224.00	0.176011
$833$	0	0
$834$	56.0000	0.00232509
$835$	19516.0	0.808837
$836$	19180.0	0.793486
$837$	3975.00	0.164153
$838$	12840.0	0.529296
$839$	29048.0	1.19529	0.597645	−	0.801761i	$-0.296103\pi$
$839$	29048.0	1.19529	0.597645	+	0.801761i	$0.296103\pi$
$840$	0	0
$841$	−13153.0	−0.539301
$842$	−20532.0	−0.840356
$843$	202.000	0.00825297
$844$	4688.00	0.191194
$845$	15113.0	0.615270
$846$	−8892.00	−0.361363
$847$	0	0
$848$	−6672.00	−0.270186
$849$	7949.00	0.321330
$850$	8968.00	0.361882
$851$	−77.0000	−0.00310168
$852$	3136.00	0.126100
$853$	32090.0	1.28809	0.644045	−	0.764988i	$-0.277255\pi$
$853$	32090.0	1.28809	0.644045	+	0.764988i	$0.277255\pi$
$854$	0	0
$855$	−24934.0	−0.997339
$856$	−1032.00	−0.0412068
$857$	−24537.0	−0.978026	−0.489013	−	0.872277i	$-0.662643\pi$
$857$	−24537.0	−0.978026	−0.489013	+	0.872277i	$0.662643\pi$
$858$	4620.00	0.183828
$859$	20825.0	0.827171	0.413585	−	0.910465i	$-0.364276\pi$
$859$	20825.0	0.827171	0.413585	+	0.910465i	$0.364276\pi$
$860$	7280.00	0.288658
$861$	0	0
$862$	30426.0	1.20222
$863$	−22847.0	−0.901183	−0.450591	−	0.892730i	$-0.648787\pi$
$863$	−22847.0	−0.901183	−0.450591	+	0.892730i	$0.648787\pi$
$864$	−1696.00	−0.0667814
$865$	11053.0	0.434466
$866$	2756.00	0.108144
$867$	1432.00	0.0560937
$868$	0	0
$869$	−17325.0	−0.676307
$870$	1484.00	0.0578302
$871$	28974.0	1.12715
$872$	7720.00	0.299808
$873$	7540.00	0.292314
$874$	1918.00	0.0742303
$875$	0	0
$876$	−1180.00	−0.0455120
$877$	−42737.0	−1.64553	−0.822763	−	0.568385i	$-0.807568\pi$
$877$	−42737.0	−1.64553	−0.822763	+	0.568385i	$0.807568\pi$
$878$	5526.00	0.212407
$879$	−318.000	−0.0122024
$880$	3920.00	0.150163
$881$	6162.00	0.235645	0.117822	−	0.993035i	$-0.462409\pi$
$881$	6162.00	0.235645	0.117822	+	0.993035i	$0.462409\pi$
$882$	0	0
$883$	7748.00	0.295290	0.147645	−	0.989040i	$-0.452831\pi$
$883$	7748.00	0.295290	0.147645	+	0.989040i	$0.452831\pi$
$884$	15576.0	0.592622
$885$	119.000	0.00451993
$886$	−11698.0	−0.443569
$887$	−25923.0	−0.981296	−0.490648	−	0.871358i	$-0.663240\pi$
$887$	−25923.0	−0.981296	−0.490648	+	0.871358i	$0.663240\pi$
$888$	88.0000	0.00332555
$889$	0	0
$890$	12222.0	0.460317
$891$	22715.0	0.854075
$892$	8096.00	0.303895
$893$	−23427.0	−0.877889
$894$	4590.00	0.171714
$895$	17157.0	0.640777
$896$	0	0
$897$	462.000	0.0171970
$898$	−9164.00	−0.340542
$899$	7950.00	0.294936
$900$	7904.00	0.292741
$901$	−24603.0	−0.909706
$902$	34860.0	1.28682
$903$	0	0
$904$	400.000	0.0147166
$905$	−8190.00	−0.300823
$906$	−2218.00	−0.0813335
$907$	31935.0	1.16911	0.584556	−	0.811353i	$-0.301269\pi$
$907$	31935.0	1.16911	0.584556	+	0.811353i	$0.301269\pi$
$908$	10284.0	0.375866
$909$	28210.0	1.02934
$910$	0	0
$911$	3408.00	0.123943	0.0619715	−	0.998078i	$-0.480261\pi$
$911$	3408.00	0.123943	0.0619715	+	0.998078i	$0.480261\pi$
$912$	−2192.00	−0.0795881
$913$	32620.0	1.18244
$914$	−23102.0	−0.836046
$915$	−357.000	−0.0128984
$916$	3580.00	0.129134
$917$	0	0
$918$	−6254.00	−0.224850
$919$	13909.0	0.499255	0.249628	−	0.968342i	$-0.419692\pi$
$919$	13909.0	0.499255	0.249628	+	0.968342i	$0.419692\pi$
$920$	392.000	0.0140477
$921$	8132.00	0.290943
$922$	18988.0	0.678239
$923$	−51744.0	−1.84526
$924$	0	0
$925$	−836.000	−0.0297162
$926$	20320.0	0.721119
$927$	−40378.0	−1.43062
$928$	−3392.00	−0.119987
$929$	−24537.0	−0.866559	−0.433279	−	0.901260i	$-0.642644\pi$
$929$	−24537.0	−0.866559	−0.433279	+	0.901260i	$0.642644\pi$
$930$	1050.00	0.0370224
$931$	0	0
$932$	7148.00	0.251224
$933$	929.000	0.0325982
$934$	2614.00	0.0915768
$935$	14455.0	0.505593
$936$	13728.0	0.479395
$937$	−32758.0	−1.14211	−0.571055	−	0.820912i	$-0.693466\pi$
$937$	−32758.0	−1.14211	−0.571055	+	0.820912i	$0.693466\pi$
$938$	0	0
$939$	209.000	0.00726353
$940$	−4788.00	−0.166135
$941$	−38561.0	−1.33587	−0.667934	−	0.744220i	$-0.732821\pi$
$941$	−38561.0	−1.33587	−0.667934	+	0.744220i	$0.732821\pi$
$942$	3118.00	0.107845
$943$	3486.00	0.120382
$944$	−272.000	−0.00937801
$945$	0	0
$946$	−18200.0	−0.625511
$947$	39661.0	1.36094	0.680470	−	0.732776i	$-0.261776\pi$
$947$	39661.0	1.36094	0.680470	+	0.732776i	$0.261776\pi$
$948$	1980.00	0.0678348
$949$	19470.0	0.665988
$950$	20824.0	0.711179
$951$	−7131.00	−0.243153
$952$	0	0
$953$	−46618.0	−1.58458	−0.792290	−	0.610144i	$-0.791111\pi$
$953$	−46618.0	−1.58458	−0.792290	+	0.610144i	$0.791111\pi$
$954$	−21684.0	−0.735897
$955$	−8925.00	−0.302415
$956$	−20400.0	−0.690150
$957$	−3710.00	−0.125316
$958$	−36574.0	−1.23346
$959$	0	0
$960$	−448.000	−0.0150616
$961$	−24166.0	−0.811185
$962$	−1452.00	−0.0486636
$963$	−3354.00	−0.112234
$964$	−16708.0	−0.558225
$965$	245.000	0.00817288
$966$	0	0
$967$	14816.0	0.492710	0.246355	−	0.969180i	$-0.420767\pi$
$967$	14816.0	0.492710	0.246355	+	0.969180i	$0.420767\pi$
$968$	848.000	0.0281568
$969$	−8083.00	−0.267970
$970$	4060.00	0.134390
$971$	−16875.0	−0.557718	−0.278859	−	0.960332i	$-0.589956\pi$
$971$	−16875.0	−0.557718	−0.278859	+	0.960332i	$0.589956\pi$
$972$	−8320.00	−0.274552
$973$	0	0
$974$	29906.0	0.983830
$975$	5016.00	0.164760
$976$	816.000	0.0267618
$977$	−15837.0	−0.518598	−0.259299	−	0.965797i	$-0.583492\pi$
$977$	−15837.0	−0.518598	−0.259299	+	0.965797i	$0.583492\pi$
$978$	−4502.00	−0.147196
$979$	−30555.0	−0.997489
$980$	0	0
$981$	25090.0	0.816577
$982$	−28704.0	−0.932771
$983$	9915.00	0.321708	0.160854	−	0.986978i	$-0.448575\pi$
$983$	9915.00	0.321708	0.160854	+	0.986978i	$0.448575\pi$
$984$	−3984.00	−0.129070
$985$	−19138.0	−0.619073
$986$	−12508.0	−0.403992
$987$	0	0
$988$	36168.0	1.16463
$989$	−1820.00	−0.0585163
$990$	12740.0	0.408994
$991$	−43681.0	−1.40017	−0.700087	−	0.714057i	$-0.746856\pi$
$991$	−43681.0	−1.40017	−0.700087	+	0.714057i	$0.746856\pi$
$992$	−2400.00	−0.0768146
$993$	6571.00	0.209994
$994$	0	0
$995$	15701.0	0.500256
$996$	−3728.00	−0.118601
$997$	−47113.0	−1.49657	−0.748287	−	0.663375i	$-0.769123\pi$
$997$	−47113.0	−1.49657	−0.748287	+	0.663375i	$0.769123\pi$
$998$	11062.0	0.350863
$999$	583.000	0.0184638

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.4.a.b.1.1		1
3.2	odd	2		882.4.a.k.1.1		1
4.3	odd	2		784.4.a.l.1.1		1
5.4	even	2		2450.4.a.bh.1.1		1
7.2	even	3		14.4.c.b.11.1	yes	2
7.3	odd	6		98.4.c.e.79.1		2
7.4	even	3		14.4.c.b.9.1	✓	2
7.5	odd	6		98.4.c.e.67.1		2
7.6	odd	2		98.4.a.c.1.1		1
21.2	odd	6		126.4.g.c.109.1		2
21.5	even	6		882.4.g.d.361.1		2
21.11	odd	6		126.4.g.c.37.1		2
21.17	even	6		882.4.g.d.667.1		2
21.20	even	2		882.4.a.p.1.1		1
28.11	odd	6		112.4.i.b.65.1		2
28.23	odd	6		112.4.i.b.81.1		2
28.27	even	2		784.4.a.j.1.1		1
35.2	odd	12		350.4.j.d.249.1		4
35.4	even	6		350.4.e.b.51.1		2
35.9	even	6		350.4.e.b.151.1		2
35.18	odd	12		350.4.j.d.149.1		4
35.23	odd	12		350.4.j.d.249.2		4
35.32	odd	12		350.4.j.d.149.2		4
35.34	odd	2		2450.4.a.bf.1.1		1
56.11	odd	6		448.4.i.d.65.1		2
56.37	even	6		448.4.i.c.193.1		2
56.51	odd	6		448.4.i.d.193.1		2
56.53	even	6		448.4.i.c.65.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.c.b.9.1	✓	2	7.4	even	3
14.4.c.b.11.1	yes	2	7.2	even	3
98.4.a.b.1.1		1	1.1	even	1	trivial
98.4.a.c.1.1		1	7.6	odd	2
98.4.c.e.67.1		2	7.5	odd	6
98.4.c.e.79.1		2	7.3	odd	6
112.4.i.b.65.1		2	28.11	odd	6
112.4.i.b.81.1		2	28.23	odd	6
126.4.g.c.37.1		2	21.11	odd	6
126.4.g.c.109.1		2	21.2	odd	6
350.4.e.b.51.1		2	35.4	even	6
350.4.e.b.151.1		2	35.9	even	6
350.4.j.d.149.1		4	35.18	odd	12
350.4.j.d.149.2		4	35.32	odd	12
350.4.j.d.249.1		4	35.2	odd	12
350.4.j.d.249.2		4	35.23	odd	12
448.4.i.c.65.1		2	56.53	even	6
448.4.i.c.193.1		2	56.37	even	6
448.4.i.d.65.1		2	56.11	odd	6
448.4.i.d.193.1		2	56.51	odd	6
784.4.a.j.1.1		1	28.27	even	2
784.4.a.l.1.1		1	4.3	odd	2
882.4.a.k.1.1		1	3.2	odd	2
882.4.a.p.1.1		1	21.20	even	2
882.4.g.d.361.1		2	21.5	even	6
882.4.g.d.667.1		2	21.17	even	6
2450.4.a.bf.1.1		1	35.34	odd	2
2450.4.a.bh.1.1		1	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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